what is the least common positive integer that meets the following conditions:
divided by 7 with remainder 4
divided by 8 with remainder 5
divided by 9 with remainder 6
i thought you could add 7 and 4 to get 13, then divide 13 and 7 with r=4, but it has to be the same number for all of them.
divisible by 9 R 6
15 24 33 42 51 60 69 78 87 96
8 remainder 5
13 21 29 37 44 52 60
7 remainder 4
11 18 25 32 39 46 53 60 ah ha
oh i see!! thank you thank you!
To find the least common positive integer that meets these conditions, we need to find the number that satisfies the given remainders for each division.
We can solve this problem using the Chinese Remainder Theorem, which states that if we have a system of linear congruences with pairwise coprime moduli, there exists a unique solution modulo the product of the moduli.
In this case, the moduli are 7, 8, and 9, which are pairwise coprime since they do not share any common factors.
To find the solution, we can use the following steps:
Step 1: Start by listing the given remainders:
Remainder 1: 4 when divided by 7
Remainder 2: 5 when divided by 8
Remainder 3: 6 when divided by 9
Step 2: Calculate the product of all moduli (7 * 8 * 9 = 504).
Step 3: For each remainder, calculate the product of the other two moduli. This is done by multiplying the other moduli together and finding the remainder when divided by the current modulus.
- For Remainder 1: 8 * 9 = 72, and the remainder when divided by 7 is 2.
- For Remainder 2: 7 * 9 = 63, and the remainder when divided by 8 is 7.
- For Remainder 3: 7 * 8 = 56, and the remainder when divided by 9 is 2.
Step 4: Calculate the product of each remainder, the product of the other two moduli, and the modular inverse of the product of the other two moduli with respect to the current modulus. This is done by finding a number y such that (product of the other two moduli * y) % current modulus = 1.
- For Remainder 1: 4 * 72 * 1 ≡ 288 ≡ 41 (mod 7).
- For Remainder 2: 5 * 63 * 7 ≡ 2205 ≡ 5 (mod 8).
- For Remainder 3: 6 * 56 * 2 ≡ 672 ≡ 7 (mod 9).
Step 5: Add up the products obtained from step 4:
(4 * 72 * 41) + (5 * 63 * 5) + (6 * 56 * 7) ≡ 108576 + 1575 + 2352 ≡ 112503 (mod 504).
Step 6: Calculate the least positive integer solution within the range of 0 to 504:
112503 % 504 ≡ 471.
Therefore, the least common positive integer that satisfies the given conditions is 471.
Note: This process can be generalized for any number of congruences and remainders.